Calculator · Structural Mechanics
Plane Stress Transformation Calculator
The same stress state looks different on differently oriented faces. Rotating the element counter-clockwise by θ redistributes the state between normal and shear components while leaving the sum σx + σy unchanged.
Stress state
σ is tensile positive. τxy acts on the +x face in the +y direction. θ is a counter-clockwise rotation of the axes in physical space.
Elements
Transformed components
| Rotation θ | 30° |
|---|---|
| σx′ | 80.981 MPa |
| σy′ | -20.981 MPa |
| τx′y′ | -28.301 MPa |
| Invariant σx + σy | 60 MPa → 60 MPa |
| Maximum in-plane shear τmax | 58.31 MPa |
Open in the Lab
The Lab sweeps θ with both elements and Mohr's circle updating together.
Technical reference
σx′ = (σx+σy)/2 + (σx−σy)/2 · cos 2θ + τxy · sin 2θ
σy′ = (σx+σy)/2 − (σx−σy)/2 · cos 2θ − τxy · sin 2θ
τx′y′ = −(σx−σy)/2 · sin 2θ + τxy · cos 2θ
- σx, σy
- normal stresses on the reference faces, tensile positive[Pa]
- τxy
- shear on the +x face acting in the +y direction[Pa]
- θ
- counter-clockwise rotation of the axes in physical space[rad]
Everything depends on 2θ, so the state repeats every 180° of physical rotation — a face and the face 180° away carry identical stresses.
σx + σy is invariant under rotation. If the two sums differ in the table above by more than rounding, the inputs are not a valid plane-stress state.
This is PLANE stress: σz = 0 on the free surface. The third principal stress is zero and is not in-plane, which matters when the maximum absolute shear is wanted rather than the maximum in-plane shear.
Assumptions
- · Plane stress: one face of the element is free, so σz = τxz = τyz = 0.
- · θ is a rotation of the observation axes, not a deformation of the material.
- · Static, linear-elastic conditions; the transformation itself is pure equilibrium and holds for any material.